Exercise 1.43: If f is a numerical function and n is a positive integer, then we can form
the nth repeated application of f, which is defined to be the function whose value at x is
f(f(...(f(x))...)). For example, if f is the function \( x \mapsto x + 1 \), then the nth
repeated application of f is the function \( x \mapsto x + n \). If f is the operation
of squaring a number, then the nth repeated application of f is the function that raises its
argument to the 2nth power. Write a procedure that takes as inputs a procedure that
computes f and a positive integer n and returns the procedure that computes the nth
repeated application of f. Your procedure should be able to be used as follows:
((repeated square 2) 5)
625
Hint: You may find it convenient to use compose from exercise 1.42.
(define (compose f g)
(lambda (x) (f (g x))))
(define (square n) (* n n))
(define (repeated fn n)
(if (= n 1)
fn
(compose fn (repeated fn (- n 1)))))
> ((repeated square 2) 5)
625
> ((repeated square 3) 5)
390625
> ((repeated (lambda (x) (+ x 1)) 3) 5)
8